A counterexample to generalizations of the Milnor-Bloch-Kato conjecture

TL;DR

This paper provides a counterexample to certain generalizations of the Milnor-Bloch-Kato conjecture by constructing specific algebraic structures where expected properties fail, challenging previous assumptions in algebraic K-theory and motives.

Contribution

The authors construct explicit counterexamples involving tori over fields that disprove the injectivity of Galois symbols and challenge generalizations of the conjecture.

Findings

Galois symbol is not injective for a specific torus over a field.

Motives of certain tori provide counterexamples to conjecture generalizations.

Challenges assumptions in algebraic K-theory and motivic conjectures.

Abstract

We construct an example of a torus $T$ over a field $K$ for which the Galois symbol $K(K; T,T)/n K(K; T,T) \to H^2(K, T[n]\otimes T[n])$ is not injective for some $n$. Here $K(K; T,T)$ is the Milnor $K$-group attached to $T$ introduced by Somekawa. We show also that the motive $M(T\times T)$ gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).